**2. Mathematical Modeling**

A two-dimensional, steady, incompressible viscous and electrically conducting nanofluid flow, comprising gyrotactic microorganisms through a stretched porous sheet by Darcy-Forchheimer relation is considered. It is also assumed that the flow field is under the e ffect of a varying magnetic field of strength *<sup>B</sup>*(*x*) = *<sup>B</sup>*0(*x*<sup>ˆ</sup>). The sheet is stretched vertically with velocity *<sup>U</sup>*<sup>0</sup>*w* = *ax*ˆ, with positive constant *a*. The induced magnetic field is ignored because it is minimal in comparison to the extraneous magnetic field, as can be seen in Figure 1. The concentration *C* 0*w*, temperature 0*Tw*, and densities for motile microorganisms are *<sup>N</sup>*<sup>0</sup>*w* and *N*0 ∞ past the stretched subsurface are considered constant and bigger than the ambient concentration *C* 0∞, temperature 0*T* ∞, respectively. It is further presumed that nanoparticles are not a ffecting the direction and velocity of microorganisms, and both the nanoparticles and the base fluid are in local thermal stability state; and the nanoparticles, motile microorganisms, and the base-fluid are having the equivalent velocities. Hence, for a suchlike problem, the governing equations for continuity, momentum, nanoparticle concentration, thermal energy, and microorganisms can be written as

$$\frac{\partial \overline{\boldsymbol{v}}}{\partial \boldsymbol{\hat{y}}} + \frac{\partial \overline{\boldsymbol{u}}}{\partial \boldsymbol{\hat{x}}} = \boldsymbol{0},\tag{1}$$

*Mathematics* **2020**, *8*, 380

$$
\begin{array}{rcl}
\widetilde{u}\frac{\partial\widetilde{u}}{\partial\widetilde{x}} + \widetilde{v}\frac{\partial\widetilde{u}}{\partial\widetilde{y}} + \sigma B\_{0}^{2}\widetilde{u} &=& -\frac{\partial\widetilde{p}}{\partial\widetilde{x}} + \nu\_{f} \Big(\frac{\partial^{2}\widetilde{u}}{\partial\widetilde{x}^{2}} + \frac{\partial^{2}\widetilde{u}}{\partial\widetilde{y}^{2}}\Big) + \widetilde{\mathbf{g}}\beta \Big(1 - \widetilde{\mathbb{C}}\_{\text{co}}\Big) \Big(\widetilde{T} - \widetilde{T}\_{\text{co}}\Big) - \widetilde{\mathbf{g}} \Big(\rho\_{p} - \rho\_{f}\Big) \Big(\widetilde{\mathbf{C}} - \widetilde{\mathbf{N}}\Big) \\
& \qquad \widetilde{\mathbf{C}}\_{\text{co}}\Big) - \widetilde{\mathbf{g}}\gamma \Big(\rho\_{m} - \rho\_{f}\Big) \Big(\widetilde{N} - \widetilde{N}\_{\text{co}}\Big) - \frac{\nu\_{f}}{k}\widetilde{u}\_{\text{r}}
\end{array} \tag{2}
$$

$$\frac{\partial \overline{\hat{p}}}{\partial \hat{y}} = 0 \tag{3}$$

$$
\begin{split}
\widetilde{\boldsymbol{u}}\frac{\partial\widetilde{\boldsymbol{T}}}{\partial\widetilde{\boldsymbol{X}}} + \widetilde{\boldsymbol{v}}\frac{\partial\widetilde{\boldsymbol{T}}}{\partial\widetilde{\boldsymbol{y}}} &= \quad \widetilde{\boldsymbol{\alpha}}\Big[\frac{\partial^{2}\widetilde{\boldsymbol{T}}}{\partial\boldsymbol{t}^{2}} + \frac{\partial^{2}\widetilde{\boldsymbol{T}}}{\partial\widetilde{\boldsymbol{y}}^{2}}\Big] + \widetilde{\boldsymbol{\tau}}\Big[\boldsymbol{D}\_{\widetilde{\mathcal{B}}}\frac{\partial\widetilde{\boldsymbol{C}}}{\partial\widetilde{\boldsymbol{y}}}\frac{\partial\widetilde{\boldsymbol{T}}}{\partial\widetilde{\boldsymbol{y}}} + \frac{\boldsymbol{D}\_{\widetilde{\mathcal{T}}}}{\widetilde{\boldsymbol{T}}\_{\rm{ov}}} \Big{{{}\left(\frac{\partial\widetilde{\boldsymbol{T}}}{\partial\widetilde{\boldsymbol{y}}}\right)}^{2} + \left(\frac{\partial\widetilde{\boldsymbol{T}}}{\partial\widetilde{\boldsymbol{x}}}\right)^{2}\Big{{}}\Big{{}\right]} + \frac{\mu\_{f}\widetilde{\boldsymbol{n}}}{k\_{t}} \Big{{}\left(\frac{\partial\widetilde{\boldsymbol{u}}}{\partial\widetilde{\boldsymbol{y}}}\right)}^{2} \\ &+ \frac{\alpha\_{f}\widetilde{\alpha}\_{0}}{k\_{l}}\widetilde{\boldsymbol{u}}^{2}
\end{split}
\tag{4}$$

$$
\overline{u}\frac{\partial\widetilde{\mathcal{C}}}{\partial\widehat{x}} + \overline{v}\frac{\partial\widetilde{\mathcal{C}}}{\partial\widehat{y}} = D\_b \left[ \frac{\partial^2\widetilde{\mathcal{C}}}{\partial\widehat{x}^2} + \frac{\partial^2\widetilde{\mathcal{C}}}{\partial\widehat{y}^2} \right] + \frac{D\_T}{T\_{\infty}} \left[ \frac{\partial^2\widetilde{T}}{\partial\widehat{x}^2} + \frac{\partial^2\widetilde{T}}{\partial\widehat{y}^2} \right] \tag{5}
$$

$$
\widetilde{\mu}\frac{\partial\widetilde{N}}{\partial\widehat{\boldsymbol{\Lambda}}} - D\_{\boldsymbol{M}} \bigg( \frac{\partial^{2}\widetilde{N}}{\partial\widehat{\boldsymbol{\Lambda}}^{2}} + \frac{\partial^{2}\widetilde{N}}{\partial\widehat{\boldsymbol{\mathcal{Y}}}^{2}} + 2\frac{\partial^{2}\widetilde{N}}{\partial\boldsymbol{\Omega}\partial\widehat{\boldsymbol{\mathcal{Y}}}} \bigg) + \widetilde{\boldsymbol{v}}\frac{\partial\widetilde{N}}{\partial\boldsymbol{\mathcal{Y}}} + \frac{b\mathcal{W}\_{\mathbb{C}}}{\left(\widetilde{\mathbb{C}} - \widetilde{\mathbb{C}}\_{\text{co}}\right)} \bigg[ \frac{\partial}{\partial\boldsymbol{\mathcal{Y}}} \bigg( \mathcal{N}\frac{\partial\widetilde{\mathbb{C}}}{\partial\boldsymbol{\mathcal{Y}}} \bigg) + \frac{\partial}{\partial\boldsymbol{\hat{\mathcal{X}}}} \bigg( \widetilde{N}\frac{\partial\widetilde{\mathbb{C}}}{\partial\boldsymbol{\hat{\mathcal{X}}}} \bigg) \bigg] = 0 \tag{6}
$$

**Figure 1.** Flow structure through a stretch elastic plate.

Their respective boundary conditions can be read as

$$
\widetilde{u} = a\mathfrak{k}, \widetilde{v} = 0, \widetilde{T} = \widetilde{T}\_w, \widetilde{\mathbb{C}} = \widetilde{\mathbb{C}}\_{w\prime}\widetilde{N} = \widetilde{N}\_w \text{ at } \widehat{\mathcal{Y}} = 0 \tag{7}
$$

$$
\overline{\mu} \to 0, \ \overline{\mathbb{C}} \to \overline{\mathbb{C}}\_{\infty}, \ \overline{\upsilon} \to 0, \ \overline{T} \to \overline{T}\_{\infty}, \ \overline{N} \to \overline{N}\_{\infty} \text{ as } \ \mathcal{G} \to \infty \tag{8}
$$

By cancelling Equation (3) from the momentum equations by cross-differentiation, only Equation (2) survives. In Equations (1)–(8), 0*u* and 0*v* are the velocity components for *x*ˆ and *y*ˆ directions correspondingly. Where 0 *T* is the temperature, *C* 0 is the concentration for nanoparticle, *N* 0 is the density for motile microorganism, 0*p* is the pressure, ρ*f* , ρ*<sup>m</sup>*, ρ*p* are the densities of nanofluid, microorganisms, and nanoparticles, *Db*, *Dm*, *DT* denote the Brownian-diffusion coefficient, diffusivity of microorganisms and thermophoresis-diffusion coefficient, *k* the porosity parameter, σ, *kt* are the electrical and thermal conductivity for the fluid, γ indicates the average volume for a microorganism, respectively. α = *kt*/ρ*cp* is the thermal diffusivity, *bWC* are the constants, and the proportion of the

effected heat capacitance of the nanoparticle to the base-fluid 0τ = (ρ*<sup>C</sup>*)*p* (ρ*<sup>C</sup>*)*f* , respectively, are the other parametric quantities.

Invoking the following transformation

$$\begin{array}{rcl} \widetilde{u} = \operatorname{af} \mathfrak{z}'(\eta), \,\widetilde{v} = \,-\sqrt{a\nu} \operatorname{g}(\eta), \,\eta = \,\sqrt{\frac{a}{\nu}} \,\mathfrak{z}'\_{\nu}\phi(\eta) & = \,\frac{\widetilde{\mathbb{C}} - \widetilde{\mathbb{C}}\_{\mathrm{ov}}}{\widetilde{\mathbb{C}}\_{w} - \widetilde{\mathbb{C}}\_{\mathrm{ov}}} \,\, \\\theta(\eta) = \,\frac{\widetilde{T} - \widetilde{T}\_{\mathrm{ov}}}{\widetilde{T}\_{w} - \widetilde{T}\_{\mathrm{ov}}} \,\,\phi(\eta) & = \,\frac{\widetilde{N} - \widetilde{N}\_{\mathrm{ov}}}{\widetilde{N}\_{w} - \widetilde{N}\_{\mathrm{ov}}} \,\, \end{array} \tag{9}$$

In Equations (1)–(8), the non-dimensional form of resulting equations, along with associated boundary conditions, can be written as

$$\log'''' + \text{gg''} - \text{g'}^2 - \text{Mg'} - \beta \text{Dg'} + \frac{G\_r}{R\_c^2} (\theta - \text{N}\_r \phi - \text{R}\_b \phi) \tag{10}$$

$$\frac{1}{P\_r}\theta^{\prime\prime} + \theta^{\prime}[\mathfrak{g} + \mathrm{N}\_b\phi^{\prime}] + \mathrm{N}\_t\theta\sigma^2 + E\_t\{\mathfrak{g}^{\prime\prime2} + \mathrm{M}\mathfrak{g}^{\prime2}\} = 0\tag{11}$$

$$
\phi'' + L\_{\mathfrak{e}} \,\phi' \,\mathcal{g} + \frac{N\_{\mathfrak{k}}}{N\_{\mathfrak{b}}} \theta'' \,\mathcal{I} \,\mathcal{J} \,\tag{12}
$$

$$\left(\phi'' + L\_b g \phi' - P\_t([\phi + \Omega\_d] \phi'' + \phi' \phi')\right) = 0 \tag{13}$$

$$\lg(\eta) = 0, \lg'(\eta) = 1, \theta(\eta) = \phi(\eta) = \phi(\eta) = 1, \text{ when } \eta = 0 \tag{14}$$

$$\text{tg}'(\eta) = 0, \theta(\eta) = \phi(\eta) = \phi(\eta) = 0, \text{ when } \eta \to \infty \tag{15}$$

In which

β*D* = ν *<sup>a</sup>*ρ*f k* , *M* = <sup>σ</sup>*B*0<sup>2</sup> *<sup>a</sup>*ρ*f* , *Gr R*2*e* = <sup>g</sup>β(<sup>1</sup>−*C*<sup>0</sup>∞)(<sup>0</sup>*T*−<sup>0</sup>*T*∞) *aU*<sup>0</sup>*w* , *Nr* = (ρ*p*<sup>−</sup>ρ*f*)(*C*<sup>0</sup>*w*<sup>−</sup>*C*<sup>0</sup>∞) βρ*f*(<sup>0</sup>*Tw*−<sup>0</sup>*T*∞)(<sup>1</sup>−*C*<sup>0</sup>∞), *Pr* = να ,*Rb* = <sup>γ</sup>(ρ*m*<sup>−</sup>ρ*f*)(*N*<sup>0</sup>*w*<sup>−</sup>*N*<sup>0</sup>∞) βρ*f*(<sup>0</sup>*Tw*−<sup>0</sup>*T*∞)(<sup>1</sup>−*C*<sup>0</sup>∞), *NT* = 0 <sup>τ</sup>*DT*(<sup>0</sup>*Tw*−<sup>0</sup>*T*∞) ν 0 *T*∞ , *Nb* = 0 <sup>τ</sup>*DB*(*C*<sup>0</sup>*w*<sup>−</sup>*C*<sup>0</sup>∞) ν , *Ec* = *U* <sup>0</sup>2 *w cp*(<sup>0</sup>*Tw*−<sup>0</sup>*T*∞), *Le* = ν*DB* , *Lb* = ν *DM* , Ω*d* = *N* 0 ∞ (*N*<sup>0</sup>*w*<sup>−</sup>*N*<sup>0</sup>∞), *Pe* = *bWC DM* , (16)

These parametric quantities are permeability parameter β*<sup>D</sup>*, Hartmann number *M*, the local Richardson number *Gr*/*R*2*e*, the buoyancy proportion parameter *Nr*, Prandtl number *Pr*, the bioconvection Rayleigh number *Rb*, the thermophoresis parameter *Nt*, the Brownian motion parameter *Nb*, Eckert number *Ec*, the conventional Lewis number and the bioconvection Lewis number *Le* and *Lb*, the bioconvection Peclet number *Pe*, and Ω*d* is the concentration of the microorganisms variance parametric quantity, respectively.

The motile density number, Sherwood, and Nusselt number for the present flow in dimensionless form are:

$$\frac{Nn\_x}{R\varepsilon\_x^{1/2}} = -\theta'(0), \frac{S\hbar\_x}{R\varepsilon\_x^{\frac{1}{2}}} = -\phi'(0), \frac{Nn\_x}{R\varepsilon\_x^{\frac{1}{2}}} = -\phi'(0),\tag{17}$$

where *Rex* = *U*0*x*<sup>ˆ</sup> ν , the local Reynolds number.
